Method and apparatus for creating consistent risk forecasts and for aggregating factor models

ABSTRACT

An invention forcing an aggregate risk model to be consistent with standalone models is provided. A revising transformation parameterized over and an objective function minimized over, the orthogonal group are provided, least changing cross blocks of covariance matrices, preserving information in original cross block correlations, consistent with a prescribed revised sub-block.

BACKGROUND OF THE INVENTION

1. Technical Field

The present invention relates to risk model aggregation. Moreparticularly, the present invention relates to a method and apparatusfor a mathematical technique that forces an aggregate risk model to beconsistent with embedded standalone models.

2. Description of the Prior Art

-   My ventures are not in one bottom trusted,-   Nor to one place; nor is my whole estate-   Upon the fortune of this present year.-   Therefore, my merchandise makes me not sad.    -   William Shakespeare, The Merchant of Venice (1598)

Shakespeare reminds us that the perils of investment have been with usalways. Investors have long appreciated the importance ofdiversification. The idea took quantitative form with the work of H.Markowitz, The Birth of Mean-Variance Optimization (7.1, 77-91),Portfolio Selection, Journal of Finance (1952). Since then increasinglysophisticated mathematical and statistical tools have been brought tobear on the problem of estimating the aggregate risk of a portfolio.

The aggregate risk of a portfolio depends crucially on the covariancesof the portfolio's constituent assets. Unfortunately, in practicalsituations so many covariances come into play that it is impossible toestimate all of them directly from historical data. Factor modelsovercome this difficulty by expressing the large number of assetcovariances in terms of a small number of factor covariances.

A factor model is defined through a linear regression, as follows:y=A·x+ε.  (1)

Here y is the vector of asset returns whose variances and covariancesrequire estimation, x is a vector of factor returns whose variances andcovariances can be reliably estimated, and A is a matrix, specified apriori, that describes the sensitivities of the assets to the factors.The vector ε of errors is usually assumed to be normally distributedwith a diagonal covariance matrix D. This model estimates the assetcovariance matrix, Σ(y), asΣ(y)≈A·Σ(x)·A ^(T) +D  (2)where Σ(x) is the matrix of factor covariances. The dimension of thematrix Σ(x) is small enough that historical data allow reasonableestimates of the covariances.

As quantitative risk management has become more sophisticated, factormodels have become more finely detailed. At the same time, models havebroadened; large models encompassing many asset classes and markets havebecome necessary for large firms to forecast their “total risk.” Both ofthese trends have forced the number of factors upward to the pointwhere, once again, more covariances are required (this time, betweenfactors instead of assets) than can be accurately estimated.

This problem can be addressed by iterating the idea that worked before.Factor models themselves can be built up from smaller factor models.However, this approach introduces a new set of difficulties related toconsistency. To illustrate the problem, imagine building a factor modelto estimate risk for a portfolio composed of US equity and fixed incomesecurities. Suppose further that there are already excellent standalonefactor models that separately treat equities and fixed incomesecurities. An aggregate factor model will almost certainly beinconsistent with the standalone factor models. The discrepancies mightresult from the iterated factor structure, from differences in theamount and frequency of data, from clashes in statistical methodsspecific to each standalone model, or from other sources. Suchdiscrepancies are undesirable in part because they can cause differentlevels of a firm to have different views of the same source ofinvestment risk.

It would be advantageous to provide a mathematical technique thatenforces consistency between an aggregate model and the standalonemodels, i.e., to achieve breadth without sacrificing meaningful detail.More specifically, it would be advantageous to revise the aggregate riskmodel to be consistent with the standalone models. Unfortunately,enforcing consistency almost always involves the destruction ofcovariance Information in the aggregate model. The problem is tominimize the damage.

SUMMARY OF THE INVENTION

The present invention relates to a method and apparatus for amathematical technique that forces an aggregate risk model to beconsistent with embedded standalone models. More specifically, thepresent invention provides an optimized method and apparatus forrevising submatrices, also referred to herein as subblocks, of acovariance matrix. A particular linear change of variables, called arevising transformation, is introduced whereby each diagonal submatrixcorresponding to a standalone model is overwritten with a revisedversion. The space of revising transformations is shown to beparameterizable by a product of orthogonal groups via a particularparameterization. A revising transformation is provided that leastchanges the cross blocks (i.e., the off-diagonal blocks) of thecovariance matrix, thereby preserving as much information in theoriginal cross block correlations as possible, consistent with aprescribed overwriting. An objective function that quantifies the lossof cross block information is specified and minimized over theorthogonal group.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of a simple matrix representation ofstocks and bonds and their respective covariance matrices according tothe invention;

FIG. 2 is a larger schematic diagram of a simplified depiction ofnumerous constituent assets and their respective covariance matricesaccording to the invention; and

FIG. 3 is a schematic diagram of a one-to-one mapping of the space ofpossible aggregate risk models that are consistent with standalonemodels 301 to the product of orthogonal groups 302 and flexibleobjective functions 303 used in the optimization process according tothe invention.

DETAILED DESCRIPTION OF THE INVENTION

The present invention relates to a method and apparatus for amathematical technique that forces an aggregate risk model to beconsistent with embedded standalone models. More specifically, thepresent invention provides an optimized method and apparatus forrevising the diagonal submatrices that correspond to standalone models,so that agreement is achieved. A particular linear change of variablesis introduced whereby each diagonal submatrix is overwritten with arevised version; the revision of a particular diagonal submatrix leavesthe others unaffected. The space of simple revising transformations isshown to be parameterizable a product of orthogonal groups via aparticular parameterization. A revising transformation is sought thatleast changes the cross blocks (Le., the off-diagonal blocks) of thecovariance matrix, thereby preserving as much of the originalinformation in the cross block correlations as possible. An objectivefunction that characterizes information loss is provided and minimizedover the product of orthogonal groups.

An important part of understanding the invention is understanding theconcept of an aggregate risk model and knowing some relevantdefinitions.

An Aggregate Risk Model and Examples

In many cases, the vector of factors underlying an aggregate risk modelcan be decomposed into meaningful subvectors. For example, a factormodel may cover more than one market or asset type. Alternatively,statistical properties may distinguish a particular group of factors.

Definitions. An aggregate factor model is a factor model whose vector offactors f decomposes into subvectors:

$\begin{matrix}{\begin{pmatrix}f_{1} \\f_{2} \\\vdots \\f_{n}\end{pmatrix}.} & (3)\end{matrix}$

Let Σ be the covariance matrix of the factors f. The decomposition inequation (3) imposes a block structure on the covariance matrix. Thei^(th) diagonal block Σ_(i) of Σ contains the subset of covariancesbetween factors in the subset f_(i). The cross-block covariances are theelements of Σ not contained in one of the Σ_(i)'s.

The problem of revising an aggregate risk model to be consistent withthe standalone models is illustrated below in two examples: a currencyblock example and a global equities model example.

A Currency Block Example

The first example arises in connection with currency risk. A substantialamount of the variation in the return of an unhedged internationalportfolio of investment-grade bonds is due to changes in currencyexchange rates. Hence international risk models need factors to accountfor currency risk, as well as risk due to changes in interest rates andcredit spreads. However, the statistical properties of the two types offactors are quite different. Currencies are very volatile and subject toregime changes. Risk estimates for these factors make use of recent,high frequency data. For local market factors, however, it is desirableto use lower frequency data and a model with a longer memory.

One approach to combining the models is to build a “first draft” of theaggregate model based on lower frequency data and then to embed atime-scaled currency risk block that has been estimated separately.Unfortunately, the easiest way to do this, i.e. simply overwriting thefirst draft of the currency block with the high frequency block, usuallyleads to an inconsistent covariance matrix. The inconsistency can arisein many ways. It might appear because the variance ascribed to a factoris too small to account for its large covariances with factors ofanother standalone model, or because it is now very strongly correlatedwith another factor from the same standalone model, but the off-diagonalblocks indicate the two factors behave quite differently in relation toa third factor from another standalone model. The “covariance matrix”resulting from the simple overwrite will characteristically makenegative variance forecasts (or imaginary forecasts of standarddeviation) for some unfortunate portfolios, and forecasts for otherportfolios will be hopelessly compromised. It follows that any realisticapproach to overwriting the currency block must disturb cross-blockcorrelations.

The preferred embodiment of the invention provides a simple approachthat produces consistency. It assumes that there is a linearrelationship between the returns forecast by the “first draft” of theaggregate model and the returns forecast by the standalone model, andtransforms from the original to the revised diagonal block followingthis relationship. In our example, a linear change of variablesoverwrites the first draft of the currency block with the high qualitystandalone model version. As is necessarily the case, this change ofvariables disturbs the cross-block correlations.

This raises the question of how small the disturbance can be made. Usinga naive linear change of variables, one can faithfully embed thecurrency block and in so doing typically change the cross-blockcorrelations by ±0.1. Using the optimized approach described below, theaverage disturbance is cut to approximately ±0.05.

A Global Equities Example

The global equities example arises in connection with a risk model for aportfolio of global equities. An example of such risk model is onecurrently in use at Barra Inc., Berkeley, Ca. Barra's global equitymodel covers 56 markets and has 1274 factors. The structural approach of“iterated factor models” mentioned above is employed. The result is a“first draft” of the risk model that avoids spurious correlations.However, it leaves a family of embedded submodels that are inconsistentwith single country equity models. Here the problem is more complicatedthan in the previous example, since many diagonal blocks requirerevision simultaneously. This results in an optimization problem withover 12,000 variables, as opposed to the roughly 700 variables in theglobal fixed income and currency embedding problem above. Thus, theoverwhelming size of the problem has led to explorations ofmodifications of the optimization routine, achieving nearly the sameresult, but with a large reduction in the number of variables. Severalsimplified approaches to multiple block embedding are discussed below inthe Section, Multiple Block Optimization.

Optimized Single Block Embedding

One preferred embodiment of the invention optimizes embedding blocks oneat a time, as if each block were the only one requiring attention. Justas in the currency/global fixed income example, the invention offers asignificant improvement over unoptimized methods.

To understand this embodiment of the invention, consider a simpleexample for vectors {tilde over (t)} and u, where ({tilde over (t)})represents, currency and (u) represents fixed income factors. Then let{tilde over (Σ)} be the covariance matrix of such random vector, {tildeover (s)}, where

$\overset{\sim}{s} = \begin{pmatrix}\overset{\sim}{t} \\u\end{pmatrix}$for vectors {tilde over (t)} and u.

This embodiment of the invention comprises overwriting a diagonalsubblock of the covariance matrix, {tilde over (Σ)} without affectingother diagonals. However, cross-blocks are affected, as shown in thelinear change of variables section below.

A Linear Change of Variables

Suppose that {tilde over (Σ)} is a covariance matrix of a random vector{tilde over (s)}. This means that {tilde over (Σ)} is symmetric andpositive semi-definite. Let L be a nonsingular linear transformationsatisfyings=L·{tilde over (s)}.  (4)Then the matrixΣ=L·{tilde over (Σ)}·L ^(T)  (5)is the covariance matrix of the random vector s. Consequently, Σ issymmetric and positive semi-definite.

We decompose the vector {tilde over (s)} of random variables into twoparts:

$\begin{matrix}{\overset{\sim}{s} = {\begin{pmatrix}\overset{\sim}{t} \\u\end{pmatrix}.}} & (6)\end{matrix}$

This breakdown induces a block structure on the covariance matrix:

$\begin{matrix}{\overset{\sim}{\Sigma}{= {\begin{pmatrix}\overset{\sim}{\Theta} & C \\C^{T} & Y\end{pmatrix}.}}} & (7)\end{matrix}$

Here {tilde over (Θ)} and Y are the covariance matrices of the vectors{tilde over (t)} and u respectively. The off-diagonal block C containscovariances between members of {tilde over (t)} and members of u. Recallthat an example to keep in mind is the breakdown into currency ({tildeover (t)}) and fixed income factors (u).

According to the preferred embodiment of the invention, the diagonalsubmatrix {tilde over (Θ)} can be overwritten with a revised version Θ.This can be accomplished with a transformation L of the following form:

$\begin{matrix}{L = {\begin{pmatrix}{\Theta^{1/2} \cdot {\overset{\sim}{\Theta}}^{{- 1}/2}} & 0 \\0 & {i\; d}\end{pmatrix}.}} & \; & (8)\end{matrix}$

The symbol {tilde over (Θ)}^(1/2) denotes the symmetric square root ofΘ. It is the unique positive semi-definite symmetric matrix M satisfyingM·M=Θ.

The symbol {tilde over (Θ)}^(−1/2) denotes the multiplicative inverse of{tilde over (Θ)}^(1/2). The matrices {tilde over (Θ)}^(1/2) and {tildeover (Θ)}^(−1/2) are guaranteed to exist, because Θ and {tilde over (Θ)}are symmetric positive semi-definite.

Due to the special form of the matrix L, transformation (5) preservesthe block structure of {tilde over (Σ)}. The matrix {tilde over (Θ)} isreplaced by Θ and Y is unchanged. The revised covariance matrix is

$\begin{matrix}\begin{pmatrix}\Theta & {\Theta^{1/2} \cdot {\overset{\sim}{\Theta}}^{{- 1}/2} \cdot C} \\{C^{T} \cdot {\overset{\sim}{\Theta}}^{{- 1}/2} \cdot \Theta^{1/2}} & Y\end{pmatrix} & \; & (9)\end{matrix}$

The considerations above show that by taking a linear transformationthat preserves the block structure of {tilde over (Σ)}, a diagonalsubblock can be overwritten without affecting the other diagonalsubblocks. However, the cross blocks are affected.

The discussion above raises two interesting issues:

-   1. Can one characterize the set of linear transformations that    revise only one diagonal block?-   2. How small can the cross-block impact of a revising transformation    be made?

These topics are addressed in the following two sections, RevisingTransformations and The Optimal Revising Transformation, respectively.

Revising Transformations

As in the previous section of the aggregate factor model, let {tildeover (Σ)} be a covariance matrix of

$\begin{matrix}{\overset{\sim}{s} = {\begin{pmatrix}\overset{\sim}{t} \\u\end{pmatrix}.}} & (10)\end{matrix}$

Let {tilde over (Θ)} be the subblock corresponding to {tilde over (t)},and let Θ be a revised version of {tilde over (Θ)}. Hence, the set oflinear transformations that revise only one diagonal block can becharacterized by a simple revising transformation, as defined below.Furthermore, the space of simple revising transformations of such set oflinear transformations can be parameterized by a particular orthogonalgroup, as shown in the discussion below.

The preferred embodiment of the invention introduces two moredefinitions.

Definition. A simple revising transformation of ({tilde over (Σ)};{tilde over (Θ)}, Θ) is a linear map L: {tilde over (s)}

s that overwrites {tilde over (Θ)} by Θ in Σ=L·{tilde over (Σ)}·L^(T)and restricts to the identity on u, so that it leaves the complementarydiagonal block unchanged.

Definition. A square root of an n×n matrix Θ is an n×n matrix √{squareroot over (Θ)} such that√{square root over (Θ)}√{square root over (Θ)}^(T)=Θ.

It is a fact that a matrix Θ has a square root if and only if Θ ispositive semi-definite symmetric. It is also not difficult to show that√{square root over (Θ)} is a square root of such Θ if and only if√{square root over (Θ)}=Θ^(1/2)·O for some orthogonal matrix O.

The preferred embodiment of the invention provides a characterization ofsimple revising transformations, presented below as Theorem 1 in twoparts, and the corresponding proof thereof.

Theorem 1

-   (a) The matrix L is a simple revising transformation of ({tilde over    (Σ)}; {tilde over (Θ)},Θ) if and only if

$L = {\begin{pmatrix}{\sqrt{\Theta}\left( \sqrt{\Theta} \right)^{- 1}} & 0 \\0 & {i\; d}\end{pmatrix}.}$for some choice of square roots √{square root over (Θ)} and √{squareroot over ({tilde over (Θ)}.

-   (b) Suppose n is the dimension of Θ. Then the space of simple    revising transformations of ({tilde over (Σ)}; {tilde over (Θ)},Θ)    is parameterized by the orthogonal group O(n) via the    parameterization

${L(O)} = \begin{pmatrix}{\Theta^{1/2} \cdot O \cdot {\overset{\sim}{\Theta}}^{{- 1}/2}} & 0 \\0 & {i\; d}\end{pmatrix}$for OεO(n).Proof of Theorem 1:

Direct calculation demonstrates that each such L or L(O) is a simplerevising transformation of ({tilde over (Σ)}; {tilde over (Θ)},Θ).

Conversely, suppose that L is a simple revising transformation of({tilde over (Σ)}; {tilde over (Θ)},Θ). By assumption L takes the form

$\begin{matrix}{L = {\begin{pmatrix}M & 0 \\0 & {i\; d}\end{pmatrix}.}} & (11)\end{matrix}$Transformation (5) implies thatΘ=M·{tilde over (Θ)}·M ^(T).  (12)

Since Θ and {tilde over (Θ)} are symmetric and positive semi-definite,equation (12) can be rewritten asid=Θ ^(−1/2) ·M·{tilde over (Θ)} ^(1/2)·{tilde over (Θ)}^(1/2) ·M^(T)··{tilde over (Θ)}^(−1/2)  (13)

Set O=Θ^(−1/2)·M·{tilde over (Θ)}^(1/2). Then equation (13) becomesid=O·O ^(T),  (14)establishing that OεO(n), and so L is of the form L(O). Since L isclearly one-to-one, this finishes the proof of (b).

For (a), if L is a simple revising transformation, then, by (b), L=L(O)for some OεO(n), and, thus, √{square root over (Θ)}=Θ^(1/2)·O and√{square root over ({tilde over (Θ)}={tilde over (Θ)}^(1/2).

The Optimal Revising Transformation

To address the second question posed in the linear change of variablessection, it is desirable to find a revising transformation that leastchanges the cross-block correlations of the covariance matrix {tildeover (Σ)}. Such a revising transformation preserves as much informationin the original cross-block correlations as possible consistent with theprescribed subblock Θ.

A preferred embodiment of the invention treats such modeling problemnumerically as a nonlinear minimization. An objective function is chosento measure the degree of change to the cross blocks due to L. Suchchoice of objective function generally depends on the relativeimportance of the different variables. It has been found that it isreasonable to make a generic choice for this analysis, e.g., theroot-mean-square change in components of the cross block due to L(O):f(O)=(1/m)∥Θ^(1/2) ·O·{tilde over (Θ)} ^(−1/2) C−C∥,where m is the number of elements of C, and ∥·∥ refers to the matrixnorm given by the square root of the sum of the squares of thecomponents.

It is equivalent and more convenient to minimize the functiong(O)=m ² f ²=∥Θ^(1/2) ·O·{tilde over (Θ)} ^(1/2) C−C∥ ².

Thus, the preferred embodiment of the invention provides the solution tothe numerical problem of minimizing g over the orthogonal group O(n). Itshould be appreciated that although this group is disconnected,attention can be restricted to the connected component containing theIdentity SO(n), the group of orthogonal matrices with determinant +1.

The form of the objective function g as a sum of squares makes itsuitable to the Levenberg-Marquardt numerical method, a hybrid ofNewton's method and steepest descent (see W. Press, et al, NumericalRecipes in C++, Cambridge, 2002). Because this method requirescomputation of derivatives of g with respect to the parameters in O, itis desirable to have a parameterization of SO(n) for which it isconvenient to compute the partial derivatives. Such a parameterizationneed not be one-to-one, but should cover the whole group.

The preferred embodiment of the invention provides such a function,defined as follows:

Let N=n(n−1)/2 denote the dimension of SO(n). For 1≦i<j≦n, denote byR_(i,j)(θ) the rotation by angle θ in the (i,j) plane (right hand rule)in R^(n). There are N distinct such coordinate planes. The preferredembodiment of the invention makes use of the following assertion.

Every element of SO(n) may be expressed as a product of N rotations,h(θ₁, . . . ,θ_(N))≡R _(1,2)(θ₁)R _(1,3)(θ₂) . . . R _(1,n)(θ_(n)) . . .R _(n−1,n)(θ_(N)),for some angles θ₁, . . . , θ_(N).

Because it is relatively easy to compute the partial derivatives of h,the Levenberg-Marquardt minimization technique is applied toG=g(h(θ₁, . . . , θ_(N)))in a straightforward way. If the method converges, the result is a valueO*εSO(n) at which g has at least a local minimum, and therefore a simplerevising transformation L(O*) likely to be better than L(id) or anotherrandom choice.Multiple Block Optimization

In the full problem, it is desirable to adjust all of the submodelsubmatrices of the first-draft covariance matrix. One approach is tosimultaneously optimize a sum of all the single block objectivefunctions. In this case, the revising transformation is a product ofsimple revising transformations.

This leads to the following result.

Theorem 2

-   (a) A revising transformation is a product of simple revising    transformations.-   (b) Let n₁, . . . , n_(k) be the dimensions of the blocks that    require revision. Then the space of revising transformation is    parameterized by the product of orthogonal groups, O(n₁)x . . .    xO(n_(k)), via the parameterization

${L\left( {O_{1},\ldots\mspace{11mu},O_{k}} \right)} = {\begin{bmatrix}{\Theta_{1}^{1/2}O_{1}{\overset{\sim}{\Theta}}_{1}^{{- 1}/2}} & \; & 0 \\\; & ⋰ & \; \\0 & \; & {\Theta_{k}^{1/2}O_{k}{\overset{\sim}{\Theta}}_{k}^{{- 1}/2}}\end{bmatrix}.}$

In practice the dimension of the product of orthogonals may be too largeto be viably searched by our algorithm. Consequently, we exploreapproaches that reduce the dimensionality of the problem.

A preferred approach is to solve a sequence of single block problems,one problem for each submodel.

FIG. 1 is a schematic diagram of a simple matrix representation ofstocks 101 and bonds 102 and their respective covariance matrices 103 aand 103 b, showing their relationship.

FIG. 2 is a larger schematic diagram of a simplified depiction ofnumerous constituent assets 201 and their respective covariance matrices202.

FIG. 3 is a schematic diagram of a one-to-one mapping of the space ofpossible aggregate risk models that are consistent with standalonesmodels 301 to the product of orthogonal groups 302 and flexibleobjective functions 303 used in the optimization process according tothe invention. It should be appreciated that the choice of objectivefunctions is flexible, not cast in stone. The objective function servesas a rating or scoring of how well the terms in the cross blocks, i.e.the cross information, the interactions, are preserved in theoptimization process. FIG. 3 shows a mock example of a point in theproduct of orthogonal groups 304 that is the best point according to theapplied objective function 303 being mapped back to a point in the spaceof all possible aggregate risk models consistent with standalone models305.

Accordingly, although the invention has been described in detail withreference to particular preferred embodiments, persons possessingordinary skill in the art to which this invention pertains willappreciate that various modifications and enhancements may be madewithout departing from the spirit and scope of the claims that follow.

1. A method for revising an aggregate risk model for a portfolio ofinvestments to be consistent with embedded standalone models, saidmethod comprising: providing a space of possible aggregate risk modelswherein each of said aggregate risk models is consistent with standalonemodels; representing said each aggregate risk model in said space as apoint in a product of orthogonal groups; providing a machine-readablemedium having stored thereon data representing sequences ofinstructions, the sequences of instructions which, when executed by aprocessor, cause the processor to perform the step of searching saidproduct of orthogonal groups for a best representative point using apredetermined objective function; and obtaining said revised aggregaterisk model consistent with standalone models by mapping said bestrepresentative point in said product of orthogonal groups back to acorresponding point in said aggregate risk model space.
 2. The method ofclaim 1, wherein best means cross blocks are minimally affected.
 3. Themethod of claim 1, wherein said predetermined objective function isflexibly chosen according to the desire of a user.
 4. A machine-readablemedium having stored thereon data representing sequences of instructionsfor revising an aggregate risk model for a portfolio of investments tobe consistent with embedded standalone models, the sequences ofinstructions which, when executed by a processor, cause the processor toperform the steps of: determining a space of possible aggregate riskmodels wherein each of said aggregate risk models is consistent withstandalone models; determining a representation of said each aggregaterisk model in said space as a point in a product of orthogonal groups;searching said product of orthogonal groups for a best representativepoint using a predetermined objective function; and determining saidrevised aggregate risk model consistent with embedded standalone modelsby mapping said best representative point in said product of orthogonalgroups back to a corresponding point in said aggregate risk model space.5. The machine-readable medium of claim 4, wherein best means crossblocks are minimally affected.
 6. A machine-readable medium havingstored thereon data representing sequences of instructions, thesequences of instructions which, when executed by a processor toparameterize by an orthogonal group O(n) a space of simple revisingtransformations ({tilde over (Σ)}; {tilde over (Θ)},Θ), each simplerevising transformation representing a risk model for a portfolio ofinvestments, via a parameterization ${L(O)} = \begin{pmatrix}{\Theta^{1/2} \cdot O \cdot {\overset{\sim}{\Theta}}^{{- 1}/2}} & 0 \\0 & {id}\end{pmatrix}$ for OεO(n), cause the processor to perform the steps of:letting a matrix L denote a simple revising transformation of said({tilde over (Σ)}; {tilde over (Θ)},Θ) if and only if$L = \begin{pmatrix}{\sqrt{\Theta}\left( \sqrt{\overset{\sim}{\Theta}} \right)^{- 1}} & 0 \\0 & {id}\end{pmatrix}$  for some choice of square roots √{square root over (Θ)}and √{square root over ({tilde over (Θ)}; letting n denote the dimensionof Θ; directly calculating to demonstrate that each such L or L(O) is asimple revising transformation of ({tilde over (Σ)}; {tilde over(Θ)},Θ); conversely, letting L denote a simple revising transformationof ({tilde over (Σ)}; {tilde over (Θ)},Θ), wherein, by assumption Ltakes the form ${L = \begin{pmatrix}M & 0 \\0 & {id}\end{pmatrix}};$  and wherein Σ=L·{tilde over (Σ)}·L^(T) implies thatΘ=M·{tilde over (Θ)}·M^(T); since Θ and {tilde over (Θ)} are symmetricand positive semi-definite, rewriting Θ=M·{tilde over (Θ)}·M^(T) asid=Θ^(−1/2)·M·{tilde over (Θ)}^(1/2)·{tilde over(Θ)}^(1/2)·M^(T)·Θ^(−1/2); setting O=Θ^(−1/2)·M·{tilde over (Θ)}^(1/2);letting id=Θ^(−1/2)·M·{tilde over (Θ)}^(1/2)·{tilde over(Θ)}^(1/2)·M^(T)·Θ^(−1/2) become id=O·O^(T), thereby establishing thatOεO(n), and so L is of the form L(O), and wherein L is clearlyone-to-one.
 7. The method of claim 6, further comprising: characterizingsimple revising transformations, said transformations revising only onediagonal block of a covariance matrix of an aggregate model by: lettinga matrix L denote a simple revising transformation of ({tilde over (Σ)};{tilde over (Θ)},Θ) if and only if $L = \begin{pmatrix}{\sqrt{\Theta}\left( \sqrt{\Theta} \right)^{- 1}} & 0 \\0 & {id}\end{pmatrix}$  for some choice of square roots √{square root over (Θ)}and √{square root over ({tilde over (Θ)}; and letting L=L(O) for someOεO(n), and, thus, √{square root over (Θ)}=Θ^(1/2)·O and √{square rootover ({tilde over (Θ)}={tilde over (Θ)}^(1/2).
 8. The method of claim 6,further comprising, for revising a plurality of subblocks of acovariance matrix: letting a revising transformation of a space ofrevising transformations be a product of said simple revisingtransformations, wherein n₁, . . . , n_(k) denote the dimensions of saidsubblocks requiring revision; and parameterizing said space of saidrevising transformations by a product of orthogonal groups, O(n₁)x . . .xO(n_(k)) via the parameterization${L\left( {O_{1},\ldots\mspace{11mu},O_{k}} \right)} = {\begin{bmatrix}{\Theta_{1}^{1/2}O_{1}{\overset{\sim}{\Theta}}_{1}^{{- 1}/2}} & \; & 0 \\\; & ⋰ & \; \\0 & \; & {\Theta_{k}^{1/2}O_{k}{\overset{\sim}{\Theta}}_{k}^{{- 1}/2}}\end{bmatrix}.}$
 9. A machine-readable medium having stored thereon datarepresenting sequences of instructions for determining a revisingtransformation, representing a risk model for a portfolio ofinvestments, that least changes the cross-blocks of a covariance matrix{tilde over (Σ)} of an aggregate risk model, wherein said revisingtransformation preserves as much information in original cross blockcorrelations as possible consistent with a prescribed subblock Θ, thesequences of instructions which, when executed by a processor, cause theprocessor to perform the steps of: choosing an objective functionƒ(O)=(1/m)∥Θ^(1/2)·O·{tilde over (Θ)}^(−1/2)C−C∥, where m is the numberof elements of C, and ∥ refers to the matrix norm given by the squareroot of the sum of the squares of the components; minimizing anequivalent and more convenient function g(O)=m²ƒ²=∥Θ^(1/2)·O·{tilde over(Θ)}^(−1/2)C−C∥²; restricting attention to the connected component oforthogonal group O(n), SO(n) containing the Identity, wherein SO(n) isthe group of orthogonal matrices with determinant +1; letting N=n(n−1)/2denote the dimension of SO(n); for 1≦i<j≦n, denoting by R_(i,j)(θ) therotation by angle θ in the (i,j) plane right hand rule in R^(n); whereinthere are N distinct such coordinate planes; expressing every element ofSO(n) as a product of N rotations, h(θ_(l), . . . ,θ_(N))≡R_(1,2)(θ₁)R_(1,3)(θ₂) . . . R_(1,n)(θ_(n)) . . .R_(n−1,n)(θ_(N)) for some angles θ₁, . . . , θ_(N); and applying aminimization technique to G=g(h(θ₁, . . . , θ_(N))); whereby if themethod converges, the result is a value O*εSO(n) at which g has at leasta local minimum, and therefore a simple revising transformation L(O*)likely to be better than L(id) or another random choice; and if themethod does not converge, then the processor stops executing thesequences of instructions.
 10. A system for creating consistent riskmodels and aggregate factor models, said system comprising: an inputdevice operable to allow entering and/or transferring input data to aprocessor, said input data representing a first transformation, saidfirst transformation representing a risk model for a portfolio ofinvestments; an output device for displaying human readable results ofmanipulation of said input data; communications buses between said inputdevice and said processor and said output device and said processor,respectively; said processor comprising a memory, wherein said memorystores a program for parameterizing by an orthogonal group O(n) a spaceof simple revising transformations ({tilde over (Σ)}; {tilde over(Θ)},Θ) via a parameterization ${L(O)} = \begin{pmatrix}{\Theta^{1/2} \cdot O \cdot {\overset{\sim}{\Theta}}^{{- 1}/2}} & 0 \\0 & {id}\end{pmatrix}$  for OεO(n), said program performing a sequence ofinstructions, the sequences of instructions, which, when executed bysaid processor, cause the processor, using said input data representingsaid first transformation, said first transformation representing saidrisk model for a portfolio of investments, to perform the step ofsearching over the orthogonal group to find a revising transformationthat minimally disturbs off-diagonal blocks; and means for displayingsaid revised transformation on said output device.
 11. The system ofclaim 10, said sequences of instructions further comprising the step of:searching over a product of orthogonal groups to find the revisingtransformation that minimally disturbs off-diagonal blocks.
 12. Thesystem of claim 10, wherein said input data and said output results arein matrix format.
 13. A method for creating consistent risk models andaggregate factor models, said method comprising: providing an inputdevice operable to allow entering and/or transferring input data to aprocessor, said input data representing a first transformation, saidfirst transformation representing a risk model for a portfolio ofinvestments; providing an output device for displaying human readableresults of manipulation of said input data; providing communicationsbuses between said input device and said processor and said outputdevice and said processor, respectively; providing a memory in saidprocessor, wherein said memory stores a program for parameterizing by anorthogonal group O(n) a space of simple revising transformations ({tildeover (Σ)}; {tilde over (Θ)},Θ) via a parameterization${L(O)} = \begin{pmatrix}{\Theta^{1/2} \cdot O \cdot {\overset{\sim}{\Theta}}^{{- 1}/2}} & 0 \\0 & {id}\end{pmatrix}$  for OεO(n), said program performing a sequence ofinstructions, the sequences of instructions, which, when executed bysaid processor, cause the processor, using said input data representingsaid first transformation, said first transformation representing saidrisk model for a portfolio of investments, to perform the step ofsearching over the orthogonal group to find a revising transformationthat minimally disturbs off-diagonal blocks; and means for displayingsaid revised transformation on said output device.
 14. The method ofclaim 13, said sequences of instructions further comprising the step of:searching over a product of orthogonal groups to find the revisingtransformation that minimally disturbs off-diagonal blocks.
 15. Themethod of claim 13, wherein said input data and said output results arein matrix format.